<s>
In	O
mathematical	O
optimization	O
,	O
Dantzig	O
's	O
simplex	B-Algorithm
algorithm	I-Algorithm
(	O
or	O
simplex	B-Algorithm
method	I-Algorithm
)	O
is	O
a	O
popular	O
algorithm	O
for	O
linear	B-Algorithm
programming	I-Algorithm
.	O
</s>
<s>
Development	O
of	O
the	O
simplex	B-Algorithm
method	I-Algorithm
was	O
evolutionary	O
and	O
happened	O
over	O
a	O
period	O
of	O
about	O
a	O
year	O
.	O
</s>
<s>
Dantzig	O
realized	O
that	O
one	O
of	O
the	O
unsolved	O
problems	O
that	O
he	O
had	O
mistaken	O
as	O
homework	O
in	O
his	O
professor	O
Jerzy	O
Neyman	O
's	O
class	O
(	O
and	O
actually	O
later	O
solved	O
)	O
,	O
was	O
applicable	O
to	O
finding	O
an	O
algorithm	O
for	O
linear	B-Algorithm
programs	I-Algorithm
.	O
</s>
<s>
This	O
problem	O
involved	O
finding	O
the	O
existence	O
of	O
Lagrange	O
multipliers	O
for	O
general	O
linear	B-Algorithm
programs	I-Algorithm
over	O
a	O
continuum	O
of	O
variables	O
,	O
each	O
bounded	O
between	O
zero	O
and	O
one	O
,	O
and	O
satisfying	O
linear	O
constraints	O
expressed	O
in	O
the	O
form	O
of	O
Lebesgue	O
integrals	O
.	O
</s>
<s>
The	O
column	O
geometry	O
used	O
in	O
this	O
thesis	O
gave	O
Dantzig	O
insight	O
that	O
made	O
him	O
believe	O
that	O
the	O
Simplex	B-Algorithm
method	I-Algorithm
would	O
be	O
very	O
efficient	O
.	O
</s>
<s>
There	O
is	O
a	O
straightforward	O
process	O
to	O
convert	O
any	O
linear	B-Algorithm
program	I-Algorithm
into	O
one	O
in	O
standard	O
form	O
,	O
so	O
using	O
this	O
form	O
of	O
linear	B-Algorithm
programs	I-Algorithm
results	O
in	O
no	O
loss	O
of	O
generality	O
.	O
</s>
<s>
An	O
extreme	O
point	O
or	O
vertex	O
of	O
this	O
polytope	O
is	O
known	O
as	O
basic	B-Algorithm
feasible	I-Algorithm
solution	I-Algorithm
(	O
BFS	O
)	O
.	O
</s>
<s>
It	O
can	O
be	O
shown	O
that	O
for	O
a	O
linear	B-Algorithm
program	I-Algorithm
in	O
standard	O
form	O
,	O
if	O
the	O
objective	O
function	O
has	O
a	O
maximum	O
value	O
on	O
the	O
feasible	O
region	O
,	O
then	O
it	O
has	O
this	O
value	O
on	O
(	O
at	O
least	O
)	O
one	O
of	O
the	O
extreme	O
points	O
.	O
</s>
<s>
This	O
in	O
itself	O
reduces	O
the	O
problem	O
to	O
a	O
finite	O
computation	O
since	O
there	O
is	O
a	O
finite	O
number	O
of	O
extreme	O
points	O
,	O
but	O
the	O
number	O
of	O
extreme	O
points	O
is	O
unmanageably	O
large	O
for	O
all	O
but	O
the	O
smallest	O
linear	B-Algorithm
programs	I-Algorithm
.	O
</s>
<s>
If	O
the	O
edge	O
is	O
finite	O
,	O
then	O
the	O
edge	O
connects	O
to	O
another	O
extreme	O
point	O
where	O
the	O
objective	O
function	O
has	O
a	O
greater	O
value	O
,	O
otherwise	O
the	O
objective	O
function	O
is	O
unbounded	O
above	O
on	O
the	O
edge	O
and	O
the	O
linear	B-Algorithm
program	I-Algorithm
has	O
no	O
solution	O
.	O
</s>
<s>
The	O
simplex	B-Algorithm
algorithm	I-Algorithm
applies	O
this	O
insight	O
by	O
walking	O
along	O
edges	O
of	O
the	O
polytope	O
to	O
extreme	O
points	O
with	O
greater	O
and	O
greater	O
objective	O
values	O
.	O
</s>
<s>
The	O
solution	O
of	O
a	O
linear	B-Algorithm
program	I-Algorithm
is	O
accomplished	O
in	O
two	O
steps	O
.	O
</s>
<s>
Depending	O
on	O
the	O
nature	O
of	O
the	O
program	O
this	O
may	O
be	O
trivial	O
,	O
but	O
in	O
general	O
it	O
can	O
be	O
solved	O
by	O
applying	O
the	O
simplex	B-Algorithm
algorithm	I-Algorithm
to	O
a	O
modified	O
version	O
of	O
the	O
original	O
program	O
.	O
</s>
<s>
The	O
possible	O
results	O
of	O
Phase	O
I	O
are	O
either	O
that	O
a	O
basic	B-Algorithm
feasible	I-Algorithm
solution	I-Algorithm
is	O
found	O
or	O
that	O
the	O
feasible	O
region	O
is	O
empty	O
.	O
</s>
<s>
In	O
the	O
latter	O
case	O
the	O
linear	B-Algorithm
program	I-Algorithm
is	O
called	O
infeasible	O
.	O
</s>
<s>
In	O
the	O
second	O
step	O
,	O
Phase	O
II	O
,	O
the	O
simplex	B-Algorithm
algorithm	I-Algorithm
is	O
applied	O
using	O
the	O
basic	B-Algorithm
feasible	I-Algorithm
solution	I-Algorithm
found	O
in	O
Phase	O
I	O
as	O
a	O
starting	O
point	O
.	O
</s>
<s>
The	O
possible	O
results	O
from	O
Phase	O
II	O
are	O
either	O
an	O
optimum	O
basic	B-Algorithm
feasible	I-Algorithm
solution	I-Algorithm
or	O
an	O
infinite	O
edge	O
on	O
which	O
the	O
objective	O
function	O
is	O
unbounded	O
above	O
.	O
</s>
<s>
The	O
transformation	O
of	O
a	O
linear	B-Algorithm
program	I-Algorithm
to	O
one	O
in	O
standard	O
form	O
may	O
be	O
accomplished	O
as	O
follows	O
.	O
</s>
<s>
The	O
second	O
equation	O
may	O
be	O
used	O
to	O
eliminate	O
from	O
the	O
linear	B-Algorithm
program	I-Algorithm
.	O
</s>
<s>
Second	O
,	O
for	O
each	O
remaining	O
inequality	O
constraint	O
,	O
a	O
new	O
variable	O
,	O
called	O
a	O
slack	B-Algorithm
variable	I-Algorithm
,	O
is	O
introduced	O
to	O
change	O
the	O
constraint	O
to	O
an	O
equality	O
constraint	O
.	O
</s>
<s>
Third	O
,	O
each	O
unrestricted	O
variable	O
is	O
eliminated	O
from	O
the	O
linear	B-Algorithm
program	I-Algorithm
.	O
</s>
<s>
The	O
equation	O
may	O
be	O
used	O
to	O
eliminate	O
from	O
the	O
linear	B-Algorithm
program	I-Algorithm
.	O
</s>
<s>
This	O
results	O
in	O
no	O
loss	O
of	O
generality	O
since	O
otherwise	O
either	O
the	O
system	O
has	O
redundant	O
equations	O
which	O
can	O
be	O
dropped	O
,	O
or	O
the	O
system	O
is	O
inconsistent	O
and	O
the	O
linear	B-Algorithm
program	I-Algorithm
has	O
no	O
solution	O
.	O
</s>
<s>
If	O
the	O
columns	O
of	O
A	O
can	O
be	O
rearranged	O
so	O
that	O
it	O
contains	O
the	O
identity	B-Algorithm
matrix	I-Algorithm
of	O
order	O
p	O
(	O
the	O
number	O
of	O
rows	O
in	O
A	O
)	O
then	O
the	O
tableau	O
is	O
said	O
to	O
be	O
in	O
canonical	O
form	O
.	O
</s>
<s>
The	O
variables	O
corresponding	O
to	O
the	O
columns	O
of	O
the	O
identity	B-Algorithm
matrix	I-Algorithm
are	O
called	O
basic	O
variables	O
while	O
the	O
remaining	O
variables	O
are	O
called	O
nonbasic	O
or	O
free	O
variables	O
.	O
</s>
<s>
If	O
the	O
values	O
of	O
the	O
nonbasic	O
variables	O
are	O
set	O
to	O
0	O
,	O
then	O
the	O
values	O
of	O
the	O
basic	O
variables	O
are	O
easily	O
obtained	O
as	O
entries	O
in	O
b	O
and	O
this	O
solution	O
is	O
a	O
basic	B-Algorithm
feasible	I-Algorithm
solution	I-Algorithm
.	O
</s>
<s>
Conversely	O
,	O
given	O
a	O
basic	B-Algorithm
feasible	I-Algorithm
solution	I-Algorithm
,	O
the	O
columns	O
corresponding	O
to	O
the	O
nonzero	O
variables	O
can	O
be	O
expanded	O
to	O
a	O
nonsingular	O
matrix	O
.	O
</s>
<s>
where	O
zB	O
is	O
the	O
value	O
of	O
the	O
objective	O
function	O
at	O
the	O
corresponding	O
basic	B-Algorithm
feasible	I-Algorithm
solution	I-Algorithm
.	O
</s>
<s>
The	O
geometrical	O
operation	O
of	O
moving	O
from	O
a	O
basic	B-Algorithm
feasible	I-Algorithm
solution	I-Algorithm
to	O
an	O
adjacent	O
basic	B-Algorithm
feasible	I-Algorithm
solution	I-Algorithm
is	O
implemented	O
as	O
a	O
pivot	O
operation	O
.	O
</s>
<s>
The	O
result	O
is	O
that	O
,	O
if	O
the	O
pivot	O
element	O
is	O
in	O
a	O
row	O
r	O
,	O
then	O
the	O
column	O
becomes	O
the	O
r-th	O
column	O
of	O
the	O
identity	B-Algorithm
matrix	I-Algorithm
.	O
</s>
<s>
The	O
variable	O
for	O
this	O
column	O
is	O
now	O
a	O
basic	O
variable	O
,	O
replacing	O
the	O
variable	O
which	O
corresponded	O
to	O
the	O
r-th	O
column	O
of	O
the	O
identity	B-Algorithm
matrix	I-Algorithm
before	O
the	O
operation	O
.	O
</s>
<s>
Let	O
a	O
linear	B-Algorithm
program	I-Algorithm
be	O
given	O
by	O
a	O
canonical	O
tableau	O
.	O
</s>
<s>
The	O
simplex	B-Algorithm
algorithm	I-Algorithm
proceeds	O
by	O
performing	O
successive	O
pivot	O
operations	O
each	O
of	O
which	O
give	O
an	O
improved	O
basic	B-Algorithm
feasible	I-Algorithm
solution	I-Algorithm
;	O
the	O
choice	O
of	O
pivot	O
element	O
at	O
each	O
step	O
is	O
largely	O
determined	O
by	O
the	O
requirement	O
that	O
this	O
pivot	O
improves	O
the	O
solution	O
.	O
</s>
<s>
If	O
there	O
is	O
more	O
than	O
one	O
column	O
so	O
that	O
the	O
entry	O
in	O
the	O
objective	O
row	O
is	O
positive	O
then	O
the	O
choice	O
of	O
which	O
one	O
to	O
add	O
to	O
the	O
set	O
of	O
basic	O
variables	O
is	O
somewhat	O
arbitrary	O
and	O
several	O
entering	O
variable	O
choice	O
rules	O
such	O
as	O
Devex	B-Algorithm
algorithm	I-Algorithm
have	O
been	O
developed	O
.	O
</s>
<s>
In	O
general	O
,	O
a	O
linear	B-Algorithm
program	I-Algorithm
will	O
not	O
be	O
given	O
in	O
the	O
canonical	O
form	O
and	O
an	O
equivalent	O
canonical	O
tableau	O
must	O
be	O
found	O
before	O
the	O
simplex	B-Algorithm
algorithm	I-Algorithm
can	O
start	O
.	O
</s>
<s>
Columns	O
of	O
the	O
identity	B-Algorithm
matrix	I-Algorithm
are	O
added	O
as	O
column	O
vectors	O
for	O
these	O
variables	O
.	O
</s>
<s>
If	O
the	O
b	O
value	O
for	O
a	O
constraint	O
equation	O
is	O
negative	O
,	O
the	O
equation	O
is	O
negated	O
before	O
adding	O
the	O
identity	B-Algorithm
matrix	I-Algorithm
columns	O
.	O
</s>
<s>
This	O
does	O
not	O
change	O
the	O
set	O
of	O
feasible	O
solutions	O
or	O
the	O
optimal	O
solution	O
,	O
and	O
it	O
ensures	O
that	O
the	O
slack	B-Algorithm
variables	I-Algorithm
will	O
constitute	O
an	O
initial	O
feasible	O
solution	O
.	O
</s>
<s>
So	O
a	O
new	O
objective	O
function	O
,	O
equal	O
to	O
the	O
sum	O
of	O
the	O
artificial	O
variables	O
,	O
is	O
introduced	O
and	O
the	O
simplex	B-Algorithm
algorithm	I-Algorithm
is	O
applied	O
to	O
find	O
the	O
minimum	O
;	O
the	O
modified	O
linear	B-Algorithm
program	I-Algorithm
is	O
called	O
the	O
Phase	O
I	O
problem	O
.	O
</s>
<s>
The	O
simplex	B-Algorithm
algorithm	I-Algorithm
applied	O
to	O
the	O
Phase	O
I	O
problem	O
must	O
terminate	O
with	O
a	O
minimum	O
value	O
for	O
the	O
new	O
objective	O
function	O
since	O
,	O
being	O
the	O
sum	O
of	O
nonnegative	O
variables	O
,	O
its	O
value	O
is	O
bounded	O
below	O
by	O
0	O
.	O
</s>
<s>
The	O
simplex	B-Algorithm
algorithm	I-Algorithm
can	O
then	O
be	O
applied	O
to	O
find	O
the	O
solution	O
;	O
this	O
step	O
is	O
called	O
Phase	O
II	O
.	O
</s>
<s>
By	O
construction	O
,	O
u	O
and	O
v	O
are	O
both	O
basic	O
variables	O
since	O
they	O
are	O
part	O
of	O
the	O
initial	O
identity	B-Algorithm
matrix	I-Algorithm
.	O
</s>
<s>
This	O
is	O
,	O
fortuitously	O
,	O
already	O
optimal	O
and	O
the	O
optimum	O
value	O
for	O
the	O
original	O
linear	B-Algorithm
program	I-Algorithm
is−	O
130/7	O
.	O
</s>
<s>
It	O
is	O
straightforward	O
to	O
avoid	O
storing	O
the	O
m	O
explicit	O
columns	O
of	O
the	O
identity	B-Algorithm
matrix	I-Algorithm
that	O
will	O
occur	O
within	O
the	O
tableau	O
by	O
virtue	O
of	O
B	O
being	O
a	O
subset	O
of	O
the	O
columns	O
of	O
 [ A , I ] 	O
.	O
</s>
<s>
This	O
implementation	O
is	O
referred	O
to	O
as	O
the	O
"	O
standard	O
simplex	B-Algorithm
algorithm	I-Algorithm
"	O
.	O
</s>
<s>
The	O
storage	O
and	O
computation	O
overhead	O
is	O
such	O
that	O
the	O
standard	O
simplex	B-Algorithm
method	I-Algorithm
is	O
a	O
prohibitively	O
expensive	O
approach	O
to	O
solving	O
large	O
linear	B-Algorithm
programming	I-Algorithm
problems	I-Algorithm
.	O
</s>
<s>
These	O
observations	O
motivate	O
the	O
"	O
revised	B-Algorithm
simplex	I-Algorithm
algorithm	I-Algorithm
"	O
,	O
for	O
which	O
implementations	O
are	O
distinguished	O
by	O
their	O
invertible	O
representation	O
ofB	O
.	O
</s>
<s>
In	O
large	O
linear-programming	B-Algorithm
problems	O
A	O
is	O
typically	O
a	O
sparse	B-Algorithm
matrix	I-Algorithm
and	O
,	O
when	O
the	O
resulting	O
sparsity	B-Algorithm
of	O
B	O
is	O
exploited	O
when	O
maintaining	O
its	O
invertible	O
representation	O
,	O
the	O
revised	B-Algorithm
simplex	I-Algorithm
algorithm	I-Algorithm
is	O
much	O
more	O
efficient	O
than	O
the	O
standard	O
simplex	B-Algorithm
method	I-Algorithm
.	O
</s>
<s>
Commercial	O
simplex	O
solvers	O
are	O
based	O
on	O
the	O
revised	B-Algorithm
simplex	I-Algorithm
algorithm	I-Algorithm
.	O
</s>
<s>
When	O
this	O
is	O
always	O
the	O
case	O
no	O
set	O
of	O
basic	O
variables	O
occurs	O
twice	O
and	O
the	O
simplex	B-Algorithm
algorithm	I-Algorithm
must	O
terminate	O
after	O
a	O
finite	O
number	O
of	O
steps	O
.	O
</s>
<s>
Basic	B-Algorithm
feasible	I-Algorithm
solutions	I-Algorithm
where	O
at	O
least	O
one	O
of	O
the	O
basic	O
variables	O
is	O
zero	O
are	O
called	O
degenerate	O
and	O
may	O
result	O
in	O
pivots	O
for	O
which	O
there	O
is	O
no	O
improvement	O
in	O
the	O
objective	O
value	O
.	O
</s>
<s>
Worse	O
than	O
stalling	O
is	O
the	O
possibility	O
the	O
same	O
set	O
of	O
basic	O
variables	O
occurs	O
twice	O
,	O
in	O
which	O
case	O
,	O
the	O
deterministic	O
pivoting	O
rules	O
of	O
the	O
simplex	B-Algorithm
algorithm	I-Algorithm
will	O
produce	O
an	O
infinite	O
loop	O
,	O
or	O
"	O
cycle	O
"	O
.	O
</s>
<s>
Bland	B-Algorithm
's	I-Algorithm
rule	I-Algorithm
prevents	O
cycling	O
and	O
thus	O
guarantees	O
that	O
the	O
simplex	B-Algorithm
algorithm	I-Algorithm
always	O
terminates	O
.	O
</s>
<s>
Another	O
pivoting	O
algorithm	O
,	O
the	O
criss-cross	B-Algorithm
algorithm	I-Algorithm
never	O
cycles	O
on	O
linear	B-Algorithm
programs	I-Algorithm
.	O
</s>
<s>
History-based	O
pivot	O
rules	O
such	O
as	O
Zadeh	B-Algorithm
's	I-Algorithm
rule	I-Algorithm
and	O
Cunningham	B-Algorithm
's	I-Algorithm
rule	I-Algorithm
also	O
try	O
to	O
circumvent	O
the	O
issue	O
of	O
stalling	O
and	O
cycling	O
by	O
keeping	O
track	O
of	O
how	O
often	O
particular	O
variables	O
are	O
being	O
used	O
and	O
then	O
favor	O
such	O
variables	O
that	O
have	O
been	O
used	O
least	O
often	O
.	O
</s>
<s>
The	O
simplex	B-Algorithm
method	I-Algorithm
is	O
remarkably	O
efficient	O
in	O
practice	O
and	O
was	O
a	O
great	O
improvement	O
over	O
earlier	O
methods	O
such	O
as	O
Fourier	B-Algorithm
–	I-Algorithm
Motzkin	I-Algorithm
elimination	I-Algorithm
.	O
</s>
<s>
However	O
,	O
in	O
1972	O
,	O
Klee	O
and	O
Minty	O
gave	O
an	O
example	O
,	O
the	O
Klee	B-Algorithm
–	I-Algorithm
Minty	I-Algorithm
cube	I-Algorithm
,	O
showing	O
that	O
the	O
worst-case	B-General_Concept
complexity	O
of	O
simplex	B-Algorithm
method	I-Algorithm
as	O
formulated	O
by	O
Dantzig	O
is	O
exponential	O
time	O
.	O
</s>
<s>
Since	O
then	O
,	O
for	O
almost	O
every	O
variation	O
on	O
the	O
method	O
,	O
it	O
has	O
been	O
shown	O
that	O
there	O
is	O
a	O
family	O
of	O
linear	B-Algorithm
programs	I-Algorithm
for	O
which	O
it	O
performs	O
badly	O
.	O
</s>
<s>
In	O
2014	O
,	O
it	O
was	O
proved	O
that	O
a	O
particular	O
variant	O
of	O
the	O
simplex	B-Algorithm
method	I-Algorithm
is	O
NP-mighty	O
,	O
i.e.	O
,	O
it	O
can	O
be	O
used	O
to	O
solve	O
,	O
with	O
polynomial	O
overhead	O
,	O
any	O
problem	O
in	O
NP	O
implicitly	O
during	O
the	O
algorithm	O
's	O
execution	O
.	O
</s>
<s>
Analyzing	O
and	O
quantifying	O
the	O
observation	O
that	O
the	O
simplex	B-Algorithm
algorithm	I-Algorithm
is	O
efficient	O
in	O
practice	O
despite	O
its	O
exponential	O
worst-case	B-General_Concept
complexity	O
has	O
led	O
to	O
the	O
development	O
of	O
other	O
measures	O
of	O
complexity	O
.	O
</s>
<s>
The	O
simplex	B-Algorithm
algorithm	I-Algorithm
has	O
polynomial-time	O
average-case	B-General_Concept
complexity	I-General_Concept
under	O
various	O
probability	O
distributions	O
,	O
with	O
the	O
precise	O
average-case	B-General_Concept
performance	O
of	O
the	O
simplex	B-Algorithm
algorithm	I-Algorithm
depending	O
on	O
the	O
choice	O
of	O
a	O
probability	O
distribution	O
for	O
the	O
random	O
matrices	O
.	O
</s>
<s>
Another	O
approach	O
to	O
studying	O
"	O
typical	O
phenomena	O
"	O
uses	O
Baire	O
category	O
theory	O
from	O
general	O
topology	O
,	O
and	O
to	O
show	O
that	O
(	O
topologically	O
)	O
"	O
most	O
"	O
matrices	O
can	O
be	O
solved	O
by	O
the	O
simplex	B-Algorithm
algorithm	I-Algorithm
in	O
a	O
polynomial	O
number	O
of	O
steps	O
.	O
</s>
<s>
Another	O
method	O
to	O
analyze	O
the	O
performance	O
of	O
the	O
simplex	B-Algorithm
algorithm	I-Algorithm
studies	O
the	O
behavior	O
of	O
worst-case	B-General_Concept
scenarios	O
under	O
small	O
perturbation	O
–	O
are	O
worst-case	B-General_Concept
scenarios	O
stable	O
under	O
a	O
small	O
change	O
(	O
in	O
the	O
sense	O
of	O
structural	O
stability	O
)	O
,	O
or	O
do	O
they	O
become	O
tractable	O
?	O
</s>
<s>
This	O
area	O
of	O
research	O
,	O
called	O
smoothed	O
analysis	O
,	O
was	O
introduced	O
specifically	O
to	O
study	O
the	O
simplex	B-Algorithm
method	I-Algorithm
.	O
</s>
<s>
Indeed	O
,	O
the	O
running	O
time	O
of	O
the	O
simplex	B-Algorithm
method	I-Algorithm
on	O
input	O
with	O
noise	O
is	O
polynomial	O
in	O
the	O
number	O
of	O
variables	O
and	O
the	O
magnitude	O
of	O
the	O
perturbations	O
.	O
</s>
<s>
Other	O
algorithms	O
for	O
solving	O
linear-programming	B-Algorithm
problems	O
are	O
described	O
in	O
the	O
linear-programming	B-Algorithm
article	O
.	O
</s>
<s>
Another	O
basis-exchange	O
pivoting	O
algorithm	O
is	O
the	O
criss-cross	B-Algorithm
algorithm	I-Algorithm
.	O
</s>
<s>
There	O
are	O
polynomial-time	O
algorithms	O
for	O
linear	B-Algorithm
programming	I-Algorithm
that	O
use	O
interior	B-Algorithm
point	I-Algorithm
methods	I-Algorithm
:	O
these	O
include	O
Khachiyan	O
's	O
ellipsoidal	B-Algorithm
algorithm	I-Algorithm
,	O
Karmarkar	O
's	O
projective	B-Algorithm
algorithm	I-Algorithm
,	O
and	O
path-following	O
algorithms	O
.	O
</s>
<s>
Linear	B-Algorithm
–	I-Algorithm
fractional	I-Algorithm
programming	I-Algorithm
(	O
LFP	O
)	O
is	O
a	O
generalization	O
of	O
linear	B-Algorithm
programming	I-Algorithm
(	O
LP	O
)	O
.	O
</s>
<s>
In	O
LP	O
the	O
objective	O
function	O
is	O
a	O
linear	B-Algorithm
function	I-Algorithm
,	O
while	O
the	O
objective	O
function	O
of	O
a	O
linear	O
–	O
fractional	O
program	O
is	O
a	O
ratio	O
of	O
two	O
linear	O
functions	O
.	O
</s>
<s>
In	O
other	O
words	O
,	O
a	O
linear	B-Algorithm
program	I-Algorithm
is	O
a	O
fractional	O
–	O
linear	B-Algorithm
program	I-Algorithm
in	O
which	O
the	O
denominator	O
is	O
the	O
constant	O
function	O
having	O
the	O
value	O
one	O
everywhere	O
.	O
</s>
<s>
A	O
linear	O
–	O
fractional	O
program	O
can	O
be	O
solved	O
by	O
a	O
variant	O
of	O
the	O
simplex	B-Algorithm
algorithm	I-Algorithm
or	O
by	O
the	O
criss-cross	B-Algorithm
algorithm	I-Algorithm
.	O
</s>
